교과서 질문
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = cos² y, y(1) = π/4
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
9장, 문제 9.3.9
5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.
y'(t) = eʸᐟ²sin t
검증된 단계별 안내1
Rewrite the given differential equation in terms of \( y \) and \( t \): \( \frac{dy}{dt} = e^{y/2} \sin t \).
Separate the variables by bringing all terms involving \( y \) to one side and all terms involving \( t \) to the other side: \( e^{-y/2} dy = \sin t \, dt \).
Integrate both sides: \( \int e^{-y/2} \, dy = \int \sin t \, dt \).
Evaluate the integrals: for the left side, use substitution to integrate \( e^{-y/2} \), and for the right side, recall that \( \int \sin t \, dt = -\cos t + C \).
After integration, solve the resulting equation explicitly for \( y \) as a function of \( t \), including the constant of integration.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Separable Differential Equations
A separable differential equation can be written as a product of a function of the dependent variable and a function of the independent variable. This allows the variables to be separated on opposite sides of the equation, enabling integration with respect to each variable independently.
추천 영상:
06:06
Solving Separable Differential Equations
Integration Techniques
Solving separable equations requires integrating both sides after separation. Familiarity with integrating exponential functions and trigonometric functions, such as sin(t), is essential to find the antiderivatives and express the solution explicitly.
추천 영상:
06:18Integration by Parts for Definite Integrals
Implicit and Explicit Solutions
After integration, solutions may be implicit or explicit. An explicit solution expresses the dependent variable directly as a function of the independent variable, which often involves algebraic manipulation or applying inverse functions to isolate the dependent variable.
추천 영상:
05:14Finding The Implicit Derivative
관련 실천
교과서 질문
What is a separable first-order differential equation?
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교과서 질문
33–42. Solving initial value problems Solve the following initial value problems.
y'(t) = 1 + eᵗ, y(0) = 4
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교과서 질문
9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.
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교과서 질문
23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.
B′(t) = 0.005B − 500, B(0) = 50,000
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교과서 질문
15–16. {Use of Tech} Solving logistic equations Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and P₀ the initial population.
r=0.2, K=300, P₀=50
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